Cosilting Modules

We study the class of modules, called cosilting modules, which are defined as the categorical duals of silting modules. Several characterizations of these modules and connections with silting modules are presented. We prove that Bazzoni’s theorem about the pure-injectivity of cotilting modules is also valid for cosilting modules.     

有限内射维数的余倾斜余模(英文)

本文中,受C.Nastasescu etc.和Y.Miyashita思想的影响,定义了余代数的余倾斜余模,研究得出有限内射维数的余倾斜余模的一些结论.     

Rings whose Injective Hulls are Cogenerators

In this paper we use hom functor to characterize rings.     

关于内射维数有限的cotilting余模

令K是域,设C是一个K-余代数,T为cotilting左C-余模,通过对cotilting余模的性质和理论的研究,得到关于cotilting余模的一些有意义的结果.     

关于余倾斜双模的k-挠自由模

Let A and F be left and right Noetherian rings and ∧ωr a cotilting bimodule. A necessary and sufficient condition for a finitely generated A-module to be ω-k-torsionfree is given and the extension closure of Tω^i is discussed. As applications, we give some results of ∧ωr related to l.id（ω） ≤ k.     

Gorenstein Cotilting and Tilting Modules

Auslander and Solberg introduced the concepts of finitely generated cotilting and tilting modules in relative homological algebra considering subfunctors of the Ext-functor. In this article we generalize Auslander–Solberg relative notions by giving the definitions of infinitely generated Gorenstein cotilting and tilting modules by means of Gorenstein exact sequences. Using the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we prove a characterization of relative Gorenstein cotilting and tilting modules, which is a generalization of the beautiful characterization of relative cotilting and tilting modules given by Bazzoni.     

凝聚环上的Cotilting 模(英)

本文首先介绍了co－*－模的概念和刻划了凝聚环的一些性质，然后刻划了凝聚环上的Cotilting模．     

Limits of Pure-Injective Cotilting Modules

The set of pure-injective cotilting modules over an artin algebra is shown to have a monoid structure. This monoid structure does not restrict down to a monoid structure on the finitely generated cotilting modules in general, but it does whenever the algebra is of finite representation type. Pure-injective cotilting modules are also constructed from any set of finitely generated cotilting modules with bounded injective dimension. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G10, 16P20, 16E30.     

Relative Projective Modules and Relative Injective Modules

Let R be a ring, and n and d fixed non-negative integers. An R-module M is called (n, d)-injective if Ext ^d+1 _R (P, M) = 0 for any n-presented R-module P. M is said to be (n, d)-projective if Ext¹ _R (M, N) = 0 for any (n, d)-injective R-module N. We use these concepts to characterize n-coherent rings and (n, d)-rings. Some known results are extended.     

内射模的t－平坦性

设Ｒ为环，ｔ是左Ｒ－模范畴的一个遗传挠理论．文中证明了下述各点等价：（１）每个内射左Ｒ－模是ｔ－平坦的；（２）每个ｔ－有限表现左Ｒ－模的内射包络是ｔ－平坦的；（３）每个ｔ－有限表现左Ｒ－模是自由Ｒ－模的子模；（４）每个ｔ－有限表现左Ｒ－模是自反的且其对偶模是Ｈ－有限生成的．     

n-Strongly Gorenstein Projective,Injective and Flat Modules

In this article, we study the relation between m-strongly Gorenstein projective (resp., injective) modules and n-strongly Gorenstein projective (resp., injective) modules whenever m ≠ n, and the homological behavior of n-strongly Gorenstein projective (resp., injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Gorenstein flat modules, and the relation between these modules and n-strongly Gorenstein projective (resp., injective) modules.     

Modules with Cosupport and Injective Functors

Several authors have studied the filtered colimit closure \varinjlimB\varinjlim\mathcal{B} of a class B\mathcal{B} of finitely presented modules. Lenzing called \varinjlimB\varinjlim\mathcal{B} the category of modules with support in B\mathcal{B}, and proved that it is equivalent to the category of flat objects in the functor category (B^op,Ab)(\mathcal{B}^\mathrm{op},\mathsf{Ab}). In this paper, we study the category (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}} of modules with cosupport in B\mathcal{B}. We show that (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}} is equivalent to the category of injective objects in (B,Ab)(\mathcal{B},\mathsf{Ab}), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hügel, Enochs, Krause, Rada, and Saorín make it easy to discuss covering and enveloping properties of (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}, and furthermore we compare the naturally associated notions of B\mathcal{B}-coherence and B\mathcal{B}-noetherianness. Finally, we prove a number of stability results for \varinjlimB\varinjlim\mathcal{B} and (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.     

Almost Projective and Almost Injective Modules

The structure of rings over which every right module is almost injective is clarified. The regular and I-finite rings over which every right module is almost projective are described.     

7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.     

On Pseudo Small Injective Modules

The definition of principally pseudo injectivity motivates us to generalize tae notion of injectivity, noted SP pseudo injectivity. The aim of this paper is to investigate characterizations and properties of SP pseudo injective modules. Various results are devel- oped, many extending known results. As applications, we give some characterizations on Noetherian rings, QI rings, quasi-Frobenius rings.     

Supplement Submodules of Injective Modules

An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R _R is F-almost injective.     

关于强n-Corenstein投射模，内射模和平坦模的注记(英文)

In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.